An independent set in a graph is a collection of vertices that are not adjacent to each other. The cardinality of the largest independent set in $G$ is represented by $\alpha(G)$. The independence polynomial of a graph $G = (V, E)$ was introduced by Gutman and Harary in 1983 and is defined as \[ I(G;x) = \sum_{k=0}^{\alpha(G)}{s_k}x^{k}={s_0}+{s_1}x+{s_2}x^{2}+...+{s_{\alpha(G)}}x^{\alpha(G)}, \] where $s_k$ represents the number of independent sets in $G$ of size $k$. The conjecture made by Alavi, Malde, Schwenk, and Erd\"os in 1987 stated that the independence polynomials of trees are unimodal, and many researchers believed that this conjecture could be strengthened up to its corresponding log-concave version. However, in our paper, we present evidence that contradicts this assumption by introducing infinite families of trees whose independence polynomials are not log-concave.

翻译：在图论中，独立集是指图中互不相邻的顶点集合。$G$中最大独立集的基数记为$\alpha(G)$。图的独立多项式由Gutman和Harary于1983年提出，定义为\[ I(G;x) = \sum_{k=0}^{\alpha(G)}{s_k}x^{k}={s_0}+{s_1}x+{s_2}x^{2}+...+{s_{\alpha(G)}}x^{\alpha(G)}, \]其中$s_k$表示$G$中大小为$k$的独立集个数。Alavi、Malde、Schwenk和Erdős于1987年提出猜想：树的独立多项式是单峰的，许多研究者认为该猜想可强化为对应的对数凹版本。然而，本文通过引入无限族树，其独立多项式并非对数凹，提供了驳斥这一假设的证据。